# Taylor expansion in Hoeffding's Lemma proof

Hoeffding's Lemma proof uses Taylor expansion with this statement:

From Taylor's theorem, for some $$0\leq \theta \leq 1$$

$$L(h) = L(0) + h L'(0) + \frac{1}{2} h^2 L''(h\theta) \leq \frac{1}{8}h^2$$

Why does it use $$0$$ in the first two terms and $$h\theta$$ in the last? But as I know they must be same in the Taylor.

This is the mean-value form of Taylor's theorem:

$$f(x)=f(0)+xf'(0)+\frac{x^2}{2}f''(c)$$ where $$c$$ is between $$0$$ and $$x$$

Take $$x=h$$ and $$c=h\theta$$

• thanks. Is it related to Lagrange error? and can you explain a good document for mean-value form of Taylor's theorem? May 27 at 9:03
• Any decent Calculus textbook.
– whuber
May 27 at 13:34
• The derivation is further down on the Wikipedia page I linked: en.wikipedia.org/wiki/… Jun 1 at 21:25